To obtain vortices in the \(x\) -\(p_y\)  plane, we need to have a long-range interaction along the \(y\)  direction, which introduces a shorter-range interaction in the \(x\) -\(p_y\)  plane.
Technology to realize long-range interactions in synthetic quantum matter has been rapidly developing; representative setups include atoms with dipole-dipole interactions [1]}, systems involving Rydberg states [2]}, [3]}, [4]}, and atoms in a cavity with light-mediated interactions [5]}.
Now we include a long-range interaction in the simplest form through the nearest-neighbor interaction \(g_1\) , which can be realized in ultracold atomic gases using dipole-dipole interactions with proper orientation of the dipole field [1]}.
In Figs. REF (c) and REF (d), we plot the ground state when \(g_s = a_{\mathrm {osc}} \omega \)  and \(g_1 = 2 a_{\mathrm {osc}} \omega \) . We see that \(|\Psi (x,p_y)|\)  has six dips. By looking at its phase profile in Fig. REF (f), we confirm that these dips are the vortices formed in the \(x\) -\(p_y\)  plane. In the \(x\) -\(y\)  plane, the state has density peaks at every other tube, as seen in the integrated density for each tube plotted in Fig. REF (e).
Vortices on a cylinder are known to have dynamical properties different from those on a plane [7]}, [8]}; our proposal provides a unique setup where such phenomena can be studied.
Since the Hamiltonian has a discrete translational symmetry along the angle \(\theta \)  from the \(x\)  axis, the state breaks this translational symmetry, thus forming a supersolid state in the \(x\) -\(y\)  plane. Whether this supersolid structure persists in the absence of the confining potential \(\tilde{\omega }\)  in a thermodynamic limit is a question to be addressed in a future work.
